# An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces

- Mouffak Benchohra
^{1}Email author, - Alberto Cabada
^{2}and - Djamila Seba
^{3}

**2009**:628916

**DOI: **10.1155/2009/628916

© The Author(s) 2009

**Received: **30 January 2009

**Accepted: **15 May 2009

**Published: **22 June 2009

## Abstract

The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of noncompactness.

## 1. Introduction

The theory of fractional differential equations has been emerging as an important area of investigation in recent years. Let us mention that this theory has many applications in describing numerous events and problems of the real world. For example, fractional differential equations are often applicable in engineering, physics, chemistry, and biology. See Hilfer[1], Glockle and Nonnenmacher [2], Metzler et al*.* [3], Podlubny [4], Gaul et al*.* [5], among others. Fractionaldifferential equations are also often an object of mathematical investigations; see the papers of Agarwal et al*.* [6], Ahmad and Nieto [7], Ahmad and Otero-Espinar [8], Belarbi et al*.* [9], Belmekki et al [10], Benchohra et al*.* [11–13], Chang and Nieto [14], Daftardar-Gejji and Bhalekar [15], Figueiredo Camargo et al. [16], and the monographs of Kilbas et al*.* [17] and Podlubny [4].

Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain , and so forth. the same requirements of boundary conditions. Caputo's fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types, see [18, 19].

where is the Caputo fractional derivative, , , and are given functions satisfying some assumptions that will be specified later, and is a Banach space with norm .

Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. Integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics [20] and cellular systems [21].

Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors such as, for instance, Arara and Benchohra [22], Benchohra et al*.* [23, 24], Infante [25], Peciulyte et al*.* [26], and the references therein.

In our investigation we apply the method associated with the technique of measures of noncompactness and the fixed point theorem of Mönch type. This technique was mainly initiated in the monograph of Bana and Goebel [27] and subsequently developed and used in many papers; see, for example, Bana and Sadarangoni [28], Guo et al. [29], Lakshmikantham and Leela [30], Mönch [31], and Szufla [32].

## 2. Preliminaries

In this section, we present some definitions and auxiliary results which will be needed in the sequel.

Let be the space of functions , whose first derivative is absolutely continuous.

Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness.

Definition 2.1 (see [27]).

Properties

The Kuratowski measure of noncompactness satisfies some properties (for more details see [27]).

(a) is compact ( is relatively compact).

(b) .

(c) .

(d)

(e)

(f) .

Here and denote the closure and the convex hull of the bounded set , respectively.

For completeness we recall the definition of Caputo derivative of fractional order.

Definition 2.2 (see [17]).

for , and as .

Here is the delta function.

Definition 2.3 (see [17]).

Here and denotes the integer part of .

Definition 2.4.

A map is said to be Carathéodory if

(i) is measurable for each

(ii) is continuous for almost each

For our purpose we will only need the following fixed point theorem and the important Lemma.

holds for every subset of , then has a fixed point.

Lemma 2.6 (see [32]).

## 3. Existence of Solutions

Let us start by defining what we mean by a solution of the problem (1.1).

Definition 3.1.

A function is said to be a solution of (1.1) if it satisfies (1.1).

Lemma 3.2 (see [11]).

if and only if is a solution of the fractional boundary value problem (3.1).

Remark 3.3.

For the forthcoming analysis, we introduce the following assumptions

(H1)The functions satisfy the Carathéodory conditions.

Theorem 3.4.

then the boundary value problem (1.1) has at least one solution.

Proof.

Clearly, the subset is closed, bounded, and convex. We will show that satisfies the assumptions of Theorem 2.5. The proof will be given in three steps.

Step 1.

is continuous.

Let be a sequence such that in . Then, for each ,

Step 2.

maps into itself.

For each , by and (3.8) we have for each

Step 3.

is bounded and equicontinuous.

By Step 2, it is obvious that is bounded.

For the equicontinuity of . Let , and . Then

As , the right-hand side of the above inequality tends to zero.

Now let be a subset of such that .

By (3.8) it follows that , that is, for each , and then is relatively compact in . In view of the Ascoli-Arzelà theorem, is relatively compact in . Applying now Theorem 2.5 we conclude that has a fixed point which is a solution of the problem (1.1).

## 4. An Example

Clearly, conditions (H1),(H2) hold with

which is satisfied for each . Then by Theorem 3.4 the problem (4.1) has a solution on .

## Declarations

### Acknowledgments

The authors thank the referees for their remarks. The research of A. Cabada has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.

## Authors’ Affiliations

## References

- Hilfer R (Ed):
*Applications of Fractional Calculus in Physics*. World Scientific, River Edge, NJ, USA; 2000:viii+463.MATHGoogle Scholar - Glockle WG, Nonnenmacher TF: A fractional calculus approach to self-similar protein dynamics.
*Biophysical Journal*1995, 68(1):46–53. 10.1016/S0006-3495(95)80157-8View ArticleGoogle Scholar - Metzler R, Schick W, Kilian H-G, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach.
*The Journal of Chemical Physics*1995, 103(16):7180–7186. 10.1063/1.470346View ArticleGoogle Scholar - Podlubny I:
*Fractional Differential Equations, Mathematics in Science and Engineering*.*Volume 198*. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.Google Scholar - Gaul L, Klein P, Kemple S: Damping description involving fractional operators.
*Mechanical Systems and Signal Processing*1991, 5(2):81–88. 10.1016/0888-3270(91)90016-XView ArticleGoogle Scholar - Agarwal RP, Benchohra M, Hamani S: Boundary value problems for differential inclusions with fractional order.
*Advanced Studies in Contemporary Mathematics*2008, 16(2):181–196.MATHMathSciNetGoogle Scholar - Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions.
*Boundary Value Problems*2009, 2009:-11.Google Scholar - Ahmad B, Otero-Espinar V: Existence of solutions for fractional differential inclusions with anti-periodic boundary conditions.
*Boundary Value Problems*2009, 2009:-11.Google Scholar - Belarbi A, Benchohra M, Ouahab A: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces.
*Applicable Analysis*2006, 85(12):1459–1470. 10.1080/00036810601066350MATHMathSciNetView ArticleGoogle Scholar - Belmekki M, Nieto JJ, Rodriguez-Lopez RR: Existence of periodic solution for a nonlinear fractional differential equation.
*Boundary Value Problems*. in press - Benchohra M, Graef JR, Hamani S: Existence results for boundary value problems with non-linear fractional differential equations.
*Applicable Analysis*2008, 87(7):851–863. 10.1080/00036810802307579MATHMathSciNetView ArticleGoogle Scholar - Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order.
*Surveys in Mathematics and Its Applications*2008, 3: 1–12.MATHMathSciNetGoogle Scholar - Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay.
*Journal of Mathematical Analysis and Applications*2008, 338(2):1340–1350. 10.1016/j.jmaa.2007.06.021MATHMathSciNetView ArticleGoogle Scholar - Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions.
*Mathematical and Computer Modelling*2009, 49(3–4):605–609. 10.1016/j.mcm.2008.03.014MATHMathSciNetView ArticleGoogle Scholar - Daftardar-Gejji V, Bhalekar S: Boundary value problems for multi-term fractional differential equations.
*Journal of Mathematical Analysis and Applications*2008, 345(2):754–765. 10.1016/j.jmaa.2008.04.065MATHMathSciNetView ArticleGoogle Scholar - Figueiredo Camargo R, Chiacchio AO, Capelas de Oliveira E: Differentiation to fractional orders and the fractional telegraph equation.
*Journal of Mathematical Physics*2008, 49(3):-12.MathSciNetView ArticleGoogle Scholar - Kilbas AA, Srivastava HM, Trujillo JJ:
*Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies*.*Volume 204*. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar - Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives.
*Rheologica Acta*2006, 45(5):765–771. 10.1007/s00397-005-0043-5View ArticleGoogle Scholar - Podlubny I: Geometric and physical interpretation of fractional integration and fractional differentiation.
*Fractional Calculus & Applied Analysis for Theory and Applications*2002, 5(4):367–386.MATHMathSciNetGoogle Scholar - Blayneh KW: Analysis of age-structured host-parasitoid model.
*Far East Journal of Dynamical Systems*2002, 4(2):125–145.MATHMathSciNetGoogle Scholar - Adomian G, Adomian GE: Cellular systems and aging models.
*Computers & Mathematics with Applications*1985, 11(1–3):283–291.MATHMathSciNetView ArticleGoogle Scholar - Arara A, Benchohra M: Fuzzy solutions for boundary value problems with integral boundary conditions.
*Acta Mathematica Universitatis Comenianae*2006, 75(1):119–126.MATHMathSciNetGoogle Scholar - Benchohra M, Hamani S, Henderson J: Functional differential inclusions with integral boundary conditions.
*Electronic Journal of Qualitative Theory of Differential Equations*2007, 2007(15):1–13.MathSciNetView ArticleGoogle Scholar - Benchohra M, Hamani S, Nieto JJ: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions. to appear in
*The Rocky Mountain Journal of Mathematics* - Infante G: Eigenvalues and positive solutions of ODEs involving integral boundary conditions.
*Discrete and Continuous Dynamical Systems*2005, 436–442.Google Scholar - Peciulyte S, Stikoniene O, Stikonas A: Sturm-Liouville problem for stationary differential operator with nonlocal integral boundary condition.
*Mathematical Modelling and Analysis*2005, 10(4):377–392.MATHMathSciNetGoogle Scholar - Banaś J, Goebel K:
*Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics*.*Volume 60*. Marcel Dekker, New York, NY, USA; 1980:vi+97.Google Scholar - Banaś J, Sadarangani K: On some measures of noncompactness in the space of continuous functions.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(2):377–383. 10.1016/j.na.2006.11.003MATHMathSciNetView ArticleGoogle Scholar - Guo D, Lakshmikantham V, Liu X:
*Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications*.*Volume 373*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:viii+341.View ArticleGoogle Scholar - Lakshmikantham V, Leela S:
*Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications*.*Volume 2*. Pergamon Press, Oxford, UK; 1981:x+258.Google Scholar - Mönch H: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*1980, 4(5):985–999. 10.1016/0362-546X(80)90010-3MATHView ArticleGoogle Scholar - Szufla S: On the application of measure of noncompactness to existence theorems.
*Rendiconti del Seminario Matematico della Università di Padova*1986, 75: 1–14.MATHMathSciNetGoogle Scholar - Agarwal RP, Meehan M, O'Regan D:
*Fixed Point Theory and Applications, Cambridge Tracts in Mathematics*.*Volume 141*. Cambridge University Press, Cambridge, UK; 2001:x+170.View ArticleGoogle Scholar

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